Large museums--by which I mean national museums--are normally dedicated to a range of materials, of which archaeology is only a small part. As a result, their collections are also large and varied; and their museum gift shops the same. While each of these museums have gift catalogs in addition to their online shops, they also have a lot of the same materials in each museum.
The Smithsonian Institution in Washington DC is one of the best ethnographic museums in the United States, and its museum store reflects its breadth of interest. Reproductions, books, desk accessories, clothing, jewelry. Online catalog orders or by phone.
The Canadian Museum of Civilization in Gatineau, Canada, is a great source for prehistory and history of Canada. The CyberBoutique has reproductions and contemporary art works from collections in the Museum, including sculptures and ceramic vessels. Also books, jewelry, games, the works. Online catalog and purchase. In French and English.
The Royal Ontario Museum in Toronto, Canada was created by the Ontario Legislature in 1912. They appear to have taken their online reproduction store offline, but they once had reproductions and "inspired pieces" from their collections, including sculpture, glass, and pottery reproductions. There's an email listing for a catalog.
MoMA has a ton of reproductions from the 1930s through the 1980s; granted, not ancient, but generally affordable. Online catalog and ordering, phone and fax.
The BM has a large collection of books and multimedia CDs about various civilizations, a children's store and some artifact replicas. Online catalog and purchase.
The Met has a collection of jewelry and sculpture among its reproductions, with a special interest in jewelry from classical Greece, Mughal India, ancient Egypt and Byzantium. Online catalog and ordering.